Benford's law, also known as the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation that in many real-life sets of numerical data, the leading digit is likely to be small. In sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. If the digits were distributed uniformly, they would each occur about 11.1% of the time. Benford's law also makes predictions about the distribution of second digits, third digits, digit combinations, and so on.
The graph to the right shows Benford's law for base 10, one of infinitely many cases of a generalized law regarding numbers expressed in arbitrary (integer) bases, which rules out the possibility that the phenomenon might be an artifact of the base-10 number system. Further generalizations published in 1995 included analogous statements for both the nth leading digit and the joint distribution of the leading n digits, the latter of which leads to a corollary wherein the significant digits are shown to be a statistically dependent quantity.
It has been shown that this result applies to a wide variety of data sets, including electricity bills, street addresses, stock prices, house prices, population numbers, death rates, lengths of rivers, and physical and mathematical constants. Like other general principles about natural data—for example, the fact that many data sets are well approximated by a normal distribution—there are illustrative examples and explanations that cover many of the cases where Benford's law applies, though there are many other cases where Benford's law applies that resist simple explanations. Benford's Law tends to be most accurate when values are distributed across multiple orders of magnitude, especially if the process generating the numbers is described by a power law (which is common in nature).
The law is similar in concept, though not identical in distribution, to Zipf's law.
The leading digits in such a set thus have the following distribution:
|d||Relative size of|
The quantity is proportional to the space between d and d + 1 on a logarithmic scale. Therefore, this is the distribution expected if the logarithms of the numbers (but not the numbers themselves) are uniformly and randomly distributed.
For example, a number x, constrained to lie between 1 and 10, starts with the digit 1 if 1 ≤ x < 2, and starts with the digit 9 if 9 ≤ x < 10. Therefore, x starts with the digit 1 if log 1 ≤ log x < log 2, or starts with 9 if log 9 ≤ log x < log 10. The interval [log 1, log 2] is much wider than the interval [log 9, log 10] (0.30 and 0.05 respectively); therefore if log x is uniformly and randomly distributed, it is much more likely to fall into the wider interval than the narrower interval, i.e. more likely to start with 1 than with 9; the probabilities are proportional to the interval widths, giving the equation above (as well as the generalization to other bases besides decimal).
Benford's law is sometimes stated in a stronger form, asserting that the fractional part of the logarithm of data is typically close to uniformly distributed between 0 and 1; from this, the main claim about the distribution of first digits can be derived.
In other bases
For b = 2, 1 (the binary and unary) number systems, Benford's law is true but trivial: All binary and unary numbers (except for 0 or the empty set) start with the digit 1. (On the other hand, the generalization of Benford's law to second and later digits is not trivial, even for binary numbers.)
Examining a list of the heights of the 58 tallest structures in the world by category shows that 1 is by far the most common leading digit, irrespective of the unit of measurement (see "scale invariance" below):
|1||24||41.4 %||16||27.6 %||30.1 %|
|2||9||15.5 %||8||13.8 %||17.6 %|
|3||7||12.1 %||5||8.6 %||12.5 %|
|4||6||10.3 %||7||12.1 %||9.7 %|
|5||1||1.7 %||10||17.2 %||7.9 %|
|6||5||8.6 %||4||6.9 %||6.7 %|
|7||1||1.7 %||2||3.4 %||5.8 %|
|8||4||6.9 %||5||8.6 %||5.1 %|
|9||1||1.7 %||1||1.7 %||4.6 %|
Another example is the leading digit of 2n. The sequence of the first 96 leading digits (1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, 1, 3, 6, 1, ... (sequence A008952 in the OEIS)) exhibits closer adherence to Benford’s law than is expected for random sequences of the same length, because it is derived from a geometric sequence.
|1||29||30.2 %||30.1 %|
|2||17||17.7 %||17.6 %|
|3||12||12.5 %||12.5 %|
|4||10||10.4 %||9.7 %|
|5||7||7.3 %||7.9 %|
|6||6||6.3 %||6.7 %|
|7||5||5.2 %||5.8 %|
|8||5||5.2 %||5.1 %|
|9||5||5.2 %||4.6 %|
The discovery of Benford's law goes back to 1881, when the Canadian-American astronomer Simon Newcomb noticed that in logarithm tables the earlier pages (that started with 1) were much more worn than the other pages. Newcomb's published result is the first known instance of this observation and includes a distribution on the second digit as well. Newcomb proposed a law that the probability of a single number N being the first digit of a number was equal to log(N + 1) − log(N).
The phenomenon was again noted in 1938 by the physicist Frank Benford, who tested it on data from 20 different domains and was credited for it. His data set included the surface areas of 335 rivers, the sizes of 3259 US populations, 104 physical constants, 1800 molecular weights, 5000 entries from a mathematical handbook, 308 numbers contained in an issue of Reader's Digest, the street addresses of the first 342 persons listed in American Men of Science and 418 death rates. The total number of observations used in the paper was 20,229. This discovery was later named after Benford (making it an example of Stigler's law).
Benford's law tends to apply most accurately to data that span several orders of magnitude. As a rule of thumb, the more orders of magnitude that the data evenly covers, the more accurately Benford's law applies. For instance, one can expect that Benford's law would apply to a list of numbers representing the populations of UK settlements. But if a "settlement" is defined as a village with population between 300 and 999, then Benford's law will not apply.
Consider the probability distributions shown below, referenced to a log scale. In each case, the total area in red is the relative probability that the first digit is 1, and the total area in blue is the relative probability that the first digit is 8. For the first distribution, the size of the areas of red and blue are approximately proportional to the widths of each red and blue bar. Therefore, the numbers drawn from this distribution will approximately follow Benford's law. On the other hand, for the second distribution, the ratio of the areas of red and blue is very different from the ratio of the widths of each red and blue bar. Rather, the relative areas of red and blue are determined more by the heights of the bars than the widths. Accordingly, the first digits in this distribution do not satisfy Benford's law at all.
Thus, real-world distributions that span several orders of magnitude rather uniformly (e.g., stock-market prices and populations of villages, towns, and cities) are likely to satisfy Benford's law very accurately. On the other hand, a distribution mostly or entirely within one order of magnitude (e.g., IQ scores or heights of human adults) is unlikely to satisfy Benford's law very accurately, if at all. However, the difference between applicable and inapplicable regimes is not a sharp cut-off: as the distribution gets narrower, the deviations from Benford's law increase gradually.
(This discussion is not a full explanation of Benford's law, because it has not explained why data sets are so often encountered that, when plotted as a probability distribution of the logarithm of the variable, are relatively uniform over several orders of magnitude.)
Krieger–Kafri entropy explanation
In 1970 Wolfgang Krieger proved what is now called the Krieger generator theorem. The Krieger generator theorem might be viewed as a justification for the assumption in the Kafri ball-and-box model that, in a given base with a fixed number of digits 0, 1, ..., n, ..., , digit n is equivalent to a Kafri box containing n non-interacting balls. A number of other scientists and statisticians have suggested entropy-related explanations for Benford's law.
Many real-world examples of Benford's law arise from multiplicative fluctuations. For example, if a stock price starts at $100, and then each day it gets multiplied by a randomly chosen factor between 0.99 and 1.01, then over an extended period the probability distribution of its price satisfies Benford's law with higher and higher accuracy.
The reason is that the logarithm of the stock price is undergoing a random walk, so over time its probability distribution will get more and more broad and smooth (see above). (More technically, the central limit theorem says that multiplying more and more random variables will create a log-normal distribution with larger and larger variance, so eventually it covers many orders of magnitude almost uniformly.) To be sure of approximate agreement with Benford's law, the distribution has to be approximately invariant when scaled up by any factor up to 10; a log-normally distributed data set with wide dispersion would have this approximate property.
Unlike multiplicative fluctuations, additive fluctuations do not lead to Benford's law: They lead instead to normal probability distributions (again by the central limit theorem), which do not satisfy Benford's law. By contrast, that hypothetical stock price described above can be written as the product of many random variables (i.e. the price change factor for each day), so is likely to follow Benford's law quite well.
Multiple probability distributions
Anton Formann provided an alternative explanation by directing attention to the interrelation between the distribution of the significant digits and the distribution of the observed variable. He showed in a simulation study that long-right-tailed distributions of a random variable are compatible with the Newcomb–Benford law, and that for distributions of the ratio of two random variables the fit generally improves. For numbers drawn from certain distributions (IQ scores, human heights) the Benford's law fails to hold because these variates obey a normal distribution, which is known not to satisfy Benford's law, since normal distributions can't span several orders of magnitude and the mantissae of their logarithms will not be (even approximately) uniformly distributed. However, if one "mixes" numbers from those distributions, for example, by taking numbers from newspaper articles, Benford's law reappears. This can also be proven mathematically: if one repeatedly "randomly" chooses a probability distribution (from an uncorrelated set) and then randomly chooses a number according to that distribution, the resulting list of numbers will obey Benford's law. A similar probabilistic explanation for the appearance of Benford's law in everyday-life numbers has been advanced by showing that it arises naturally when one considers mixtures of uniform distributions.
In a list of lengths, the distribution of first digits of numbers in the list may be generally similar regardless of whether all the lengths are expressed in metres, yards, feet, inches, etc. The same applies to monetary units.
This is not always the case. For example, the height of adult humans almost always starts with a 1 or 2 when measured in metres and almost always starts with 4, 5, 6, or 7 when measured in feet. But in a list of lengths spread evenly over many orders of magnitude—for example, a list of 1000 lengths mentioned in scientific papers that includes the measurements of molecules, bacteria, plants, and galaxies—it is reasonable to expect the distribution of first digits to be the same no matter whether the lengths are written in metres or in feet.
For example, the first (non-zero) digit on the aforementioned list of lengths should have the same distribution whether the unit of measurement is feet or yards. But there are three feet in a yard, so the probability that the first digit of a length in yards is 1 must be the same as the probability that the first digit of a length in feet is 3, 4, or 5; similarly, the probability that the first digit of a length in yards is 2 must be the same as the probability that the first digit of a length in feet is 6, 7, or 8. Applying this to all possible measurement scales gives the logarithmic distribution of Benford's law.
Accounting fraud detection
In 1972, Hal Varian suggested that the law could be used to detect possible fraud in lists of socio-economic data submitted in support of public planning decisions. Based on the plausible assumption that people who fabricate figures tend to distribute their digits fairly uniformly, a simple comparison of first-digit frequency distribution from the data with the expected distribution according to Benford's law ought to show up any anomalous results.
Use in criminal trials
In the United States, evidence based on Benford's law has been admitted in criminal cases at the federal, state, and local levels.
Walter Mebane, a political scientist and statistician at the University of Michigan, was the first to apply the second-digit Benford's law-test (2BL-test) in election forensics. Such analysis is considered a simple, though not foolproof, method of identifying irregularities in election results. Scientific consensus to support the applicability of Benford's law to elections has not been reached in the literature. A 2011 study by the political scientists Joseph Deckert, Mikhail Myagkov, and Peter C. Ordeshook argued that Benford's law is problematic and misleading as a statistical indicator of election fraud. Their method was criticized by Mebane in a response, though he agreed that there are many caveats to the application of Benford's law to election data.
Benford's law has been used as evidence of fraud in the 2009 Iranian elections. An analysis by Mebane found that the second digits in vote counts for President Mahmoud Ahmadinejad, the winner of the election, tended to differ significantly from the expectations of Benford's law, and that the ballot boxes with very few invalid ballots had a greater influence on the results, suggesting widespread ballot stuffing. Another study used bootstrap simulations to find that the candidate Mehdi Karroubi received almost twice as many vote counts beginning with the digit 7 as would be expected according to Benford's law, while an analysis from Columbia University concluded that the probability that a fair election would produce both too few non-adjacent digits and the suspicious deviations in last-digit frequencies as found in the 2009 Iranian presidential election is less than 0.5 percent. Benford's law has also been applied for forensic auditing and fraud detection on data from the 2003 California gubernatorial election, the 2000 and 2004 United States presidential elections, and the 2009 German federal election; the Benford's Law Test was found to be "worth taking seriously as a statistical test for fraud," although "is not sensitive to distortions we know significantly affected many votes."[further explanation needed]
Benford's law has also been misapplied to claim election fraud. When applying the law to Joe Biden's election returns for Chicago, Milwaukee, and other localities in the 2020 United States presidential election, the distribution of the first digit did not follow Benford's law. The misapplication was a result of looking at data that was tightly bound in range, which violates the assumption inherent in Benford's law that the range of the data be large. The first digit test was applied to precinct-level data, but because precincts rarely receive more than a few thousand votes or fewer than several dozen, Benford's law cannot be expected to apply. According to Mebane, "It is widely understood that the first digits of precinct vote counts are not useful for trying to diagnose election frauds."
Similarly, the macroeconomic data the Greek government reported to the European Union before entering the eurozone was shown to be probably fraudulent using Benford's law, albeit years after the country joined.
Price digit analysis
Benford's law as a benchmark for the investigation of price digits has been successfully introduced into the context of pricing research. The importance of this benchmark for detecting irregularities in prices was first demonstrated in a Europe-wide study which investigated consumer price digits before and after the euro introduction for price adjustments. The introduction of the euro in 2002, with its various exchange rates, distorted existing nominal price patterns while at the same time retaining real prices. While the first digits of nominal prices distributed according to Benford's law, the study showed a clear deviation from this benchmark for the second and third digits in nominal market prices with a clear trend towards psychological pricing after the nominal shock of the euro introduction.
The number of open reading frames and their relationship to genome size differs between eukaryotes and prokaryotes with the former showing a log-linear relationship and the latter a linear relationship. Benford's law has been used to test this observation with an excellent fit to the data in both cases.
Scientific fraud detection
A test of regression coefficients in published papers showed agreement with Benford's law. As a comparison group subjects were asked to fabricate statistical estimates. The fabricated results conformed to Benford's law on first digits, but failed to obey Benford's law on second digits.
Although the chi-squared test has been used to test for compliance with Benford's law it has low statistical power when used with small samples.
The Kolmogorov–Smirnov test and the Kuiper test are more powerful when the sample size is small, particularly when Stephens's corrective factor is used. These tests may be unduly conservative when applied to discrete distributions. Values for the Benford test have been generated by Morrow. The critical values of the test statistics are shown below:
0.10 0.05 0.01 Kuiper 1.191 1.321 1.579 Kolmogorov–Smirnov 1.012 1.148 1.420
These critical values provide the minimum test statistic values required to reject the hypothesis of compliance with Benford's law at the given significance levels.
Two alternative tests specific to this law have been published: First, the max (m) statistic is given by
Secondly, the distance (d) statistic is given by
where FSD is the first significant digit and N is the sample size. Morrow has determined the critical values for both these statistics, which are shown below:
0.10 0.05 0.01 Leemis's m 0.851 0.967 1.212 Cho & Gaines's d 1.212 1.330 1.569
Morrow has also shown that for any random variable X (with a continuous PDF) divided by its standard deviation (σ), some value A can be found so that the probability of the distribution of the first significant digit of the random variable will differ from Benford's law by less than ε > 0. The value of A depends on the value of ε and the distribution of the random variable.
A method of accounting fraud detection based on bootstrapping and regression has been proposed.
If the goal is to conclude agreement with the Benford's law rather than disagreement, then the goodness-of-fit tests mentioned above are inappropriate. In this case the specific tests for equivalence should be applied. An empirical distribution is called equivalent to the Benford's law if a distance (for example total variation distance or the usual Euclidean distance) between the probability mass functions is sufficiently small. This method of testing with application to Benford's law is described in Ostrovski.
Range of applicability
Distributions known to obey Benford's law
Some well-known infinite integer sequences provably satisfy Benford's law exactly (in the asymptotic limit as more and more terms of the sequence are included). Among these are the Fibonacci numbers, the factorials, the powers of 2, and the powers of almost any other number.
Likewise, some continuous processes satisfy Benford's law exactly (in the asymptotic limit as the process continues through time). One is an exponential growth or decay process: If a quantity is exponentially increasing or decreasing in time, then the percentage of time that it has each first digit satisfies Benford's law asymptotically (i.e. increasing accuracy as the process continues through time).
Distributions known to disobey Benford's law
The square roots and reciprocals of successive natural numbers do not obey this law. Prime numbers in a finite range follow a Generalized Benford’s law, that approaches uniformity as the size of the range approaches infinity. Lists of local telephone numbers violate Benford's law. Benford's law is violated by the populations of all places with a population of at least 2500 individuals from five US states according to the 1960 and 1970 censuses, where only 19 % began with digit 1 but 20 % began with digit 2, because truncation at 2500 introduces statistical bias. The terminal digits in pathology reports violate Benford's law due to rounding.
Criteria for distributions expected and not expected to obey Benford's law
A number of criteria, applicable particularly to accounting data, have been suggested where Benford's law can be expected to apply.
- Distributions that can be expected to obey Benford's law
- When the mean is greater than the median and the skew is positive
- Numbers that result from mathematical combination of numbers: e.g. quantity × price
- Transaction level data: e.g. disbursements, sales
- Distributions that would not be expected to obey Benford's law
- Where numbers are assigned sequentially: e.g. check numbers, invoice numbers
- Where numbers are influenced by human thought: e.g. prices set by psychological thresholds ($9.99)
- Accounts with a large number of firm-specific numbers: e.g. accounts set up to record $100 refunds
- Accounts with a built-in minimum or maximum
- Distributions that do not span an order of magnitude of numbers.
Benford’s law compliance theorem
Mathematically, Benford’s law applies if the distribution being tested fits the "Benford’s law compliance theorem". The derivation says that Benford's law is followed if the Fourier transform of the logarithm of the probability density function is zero for all integer values. Most notably, this is satisfied if the Fourier transform is zero (or negligible) for n ≥ 1. This is satisfied if the distribution is wide (since wide distribution implies a narrow Fourier transform). Smith summarizes thus (p. 716):
Benford's law is followed by distributions that are wide compared with unit distance along the logarithmic scale. Likewise, the law is not followed by distributions that are narrow compared with unit distance … If the distribution is wide compared with unit distance on the log axis, it means that the spread in the set of numbers being examined is much greater than ten.
In short, Benford’s law requires that the numbers in the distribution being measured have a spread across at least an order of magnitude.
Tests with common distributions
Benford's law was empirically tested against the numbers (up to the 10th digit) generated by a number of important distributions, including the uniform distribution, the exponential distribution, the normal distribution, and others.
Neither the normal distribution nor the ratio distribution of two normal distributions (the Cauchy distribution) obey Benford's law. Although the half-normal distribution does not obey Benford's law, the ratio distribution of two half-normal distributions does. Neither the right-truncated normal distribution nor the ratio distribution of two right-truncated normal distributions are well described by Benford's law. This is not surprising as this distribution is weighted towards larger numbers.
Benford's law also describes the exponential distribution and the ratio distribution of two exponential distributions well. The fit of chi-squared distribution depends on the degrees of freedom (df) with good agreement with df = 1 and decreasing agreement as the df increases. The F-distribution is fitted well for low degrees of freedom. With increasing dfs the fit decreases but much more slowly than the chi-squared distribution. The fit of the log-normal distribution depends on the mean and the variance of the distribution. The variance has a much greater effect on the fit than does the mean. Larger values of both parameters result in better agreement with the law. The ratio of two log normal distributions is a log normal so this distribution was not examined.
Other distributions that have been examined include the Muth distribution, Gompertz distribution, Weibull distribution, gamma distribution, log-logistic distribution and the exponential power distribution all of which show reasonable agreement with the law. The Gumbel distribution – a density increases with increasing value of the random variable – does not show agreement with this law.
Generalization to digits beyond the first
It is possible to extend the law to digits beyond the first. In particular, for any given number of digits, the probability of encountering a number starting with the string of digits n of that length – discarding leading zeros – is given by
For example, the probability that a number starts with the digits 3, 1, 4 is log10(1 + 1/314) ≈ 0.00138, as in the figure on the right. Numbers satisfying this include 3.14159..., 314285.7... and 0.00314465... .
This result can be used to find the probability that a particular digit occurs at a given position within a number. For instance, the probability that a "2" is encountered as the second digit is
And the probability that d (d = 0, 1, ..., 9) is encountered as the n-th (n > 1) digit is
The distribution of the n-th digit, as n increases, rapidly approaches a uniform distribution with 10% for each of the ten digits, as shown below. Four digits is often enough to assume a uniform distribution of 10% as "0" appears 10.0176% of the time in the fourth digit, while "9" appears 9.9824% of the time.
For the two-digit distribution according to Benford's law these values are also known:
A table of the exact probabilities for the joint occurrence of the first two digits according to Benford's law is available, as is the population correlation between the first and second digits: ρ = 0.0561.
In popular culture
Benford's law has appeared as a plot device in some twenty-first century popular entertainment.
- Television crime drama NUMB3RS used Benford's law in the 2006 episode "The Running Man" to help solve a series of high burglaries.
- The 2016 film The Accountant relied on Benford's law to expose theft of funds from a robotics company.
- The 2017 Netflix series Ozark used Benford's law to analyze a cartel member's financial statements and uncover fraud.
- The 2021 Jeremy Robinson novel Infinite 2 applied Benford's law to test whether the characters are in a simulation or reality.
- Arno Berger and Theodore P. Hill, Benford's Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem, 2011.
- Weisstein, Eric W. "Benford's Law". MathWorld, A Wolfram web resource. Retrieved 7 June 2015.
- Hill, Theodore (1995). "A Statistical Derivation of the Significant-Digit Law". Statistical Science. 10 (4). doi:10.1214/ss/1177009869.
- Paul H. Kvam, Brani Vidakovic, Nonparametric Statistics with Applications to Science and Engineering, p. 158.
- Berger, Arno; Hill, Theodore P. (30 June 2020). "The mathematics of Benford's law: a primer". Stat. Methods Appl. 30 (3): 779–795. arXiv:1909.07527. doi:10.1007/s10260-020-00532-8. S2CID 202583554.
- Cai, Zhaodong; Faust, Matthew; Hildebrand, A. J.; Li, Junxian; Zhang, Yuan (15 March 2020). "The Surprising Accuracy of Benford's Law in Mathematics". The American Mathematical Monthly. 127 (3): 217–237. arXiv:1907.08894. doi:10.1080/00029890.2020.1690387. ISSN 0002-9890. S2CID 198147766.
- Frank Benford (March 1938). "The law of anomalous numbers". Proc. Am. Philos. Soc. 78 (4): 551–572. JSTOR 984802.
- Simon Newcomb (1881). "Note on the frequency of use of the different digits in natural numbers". American Journal of Mathematics. 4 (1/4): 39–40. Bibcode:1881AmJM....4...39N. doi:10.2307/2369148. JSTOR 2369148. S2CID 124556624.
- Formann, A. K. (2010). Morris, Richard James (ed.). "The Newcomb–Benford Law in Its Relation to Some Common Distributions". PLOS ONE. 5 (5): e10541. Bibcode:2010PLoSO...510541F. doi:10.1371/journal.pone.0010541. PMC 2866333. PMID 20479878.
- Miller, Steven J., ed. (9 June 2015). Benford's Law: Theory and Applications. Princeton University Press. p. 309. ISBN 978-1-4008-6659-5.
- They should strictly be bars but are shown as lines for clarity.
- Pimbley, J. M. (2014). "Benford's Law as a Logarithmic Transformation" (PDF). Maxwell Consulting, LLC. Archived (PDF) from the original on 9 October 2022. Retrieved 15 November 2020.
- Khosravani, A. (2012). Transformation Invariance of Benford Variables and their Numerical Modeling. Recent Researches in Automatic Control and Electronics. pp. 57–61. ISBN 978-1-61804-080-0.
- That the first 100 powers of 2 approximately satisfy Benford's law is mentioned by Ralph Raimi. Raimi, Ralph A. (1976). "The First Digit Problem". American Mathematical Monthly. 83 (7): 521–538. doi:10.2307/2319349. JSTOR 2319349.
- Theodore P. Hill (1995). "A Statistical Derivation of the Significant-Digit Law". Statistical Science. 10 (4): 354–363. doi:10.1214/ss/1177009869. MR 1421567.
- Hill, Theodore P. (1995). "Base-invariance implies Benford's law". Proceedings of the American Mathematical Society. 123 (3): 887–895. doi:10.1090/S0002-9939-1995-1233974-8. ISSN 0002-9939.
- Steven W. Smith. "Chapter 34: Explaining Benford's Law. The Power of Signal Processing". The Scientist and Engineer's Guide to Digital Signal Processing. Retrieved 15 December 2012.
- Fewster, R. M. (2009). "A simple explanation of Benford's Law" (PDF). The American Statistician. 63 (1): 26–32. CiteSeerX 10.1.1.572.6719. doi:10.1198/tast.2009.0005. S2CID 39595550. Archived (PDF) from the original on 9 October 2022.
- Arno Berger and Theodore P. Hill, Benford's Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem, 2011. The authors describe this argument but say it "still leaves open the question of why it is reasonable to assume that the logarithm of the spread, as opposed to the spread itself—or, say, the log log spread—should be large" and that "assuming large spread on a logarithmic scale is equivalent to assuming an approximate conformance with [Benford's law]" (italics added), something which they say lacks a "simple explanation".
- Krieger, Wolfgang (1970). "On entropy and generators of measure-preserving transformations". Transactions of the American Mathematical Society. 149 (2): 453. doi:10.1090/S0002-9947-1970-0259068-3. ISSN 0002-9947.
- Downarowicz, Tomasz (12 May 2011). Entropy in Dynamical Systems. Cambridge University Press. p. 106. ISBN 978-1-139-50087-6.
- Smorodinsky, Meir (1971). "Chapter IX. Entropy and generators. Krieger's theorem". Ergodic Theory, Entropy. Lecture Notes in Mathematics. Vol. 214. Berlin, Heidelberg: Springer. doi:10.1007/BFb0066096.
- Jolion, Jean-Michel (2001). "Images and Benford's Law". Journal of Mathematical Imaging and Vision. 14 (1): 73–81. doi:10.1023/A:1008363415314. ISSN 0924-9907. S2CID 34151059.
- Lemons, Don S. (2019). "Thermodynamics of Benford's first digit law". American Journal of Physics. 87 (10): 787–790. arXiv:1604.05715. Bibcode:2019AmJPh..87..787L. doi:10.1119/1.5116005. ISSN 0002-9505. S2CID 119207367.
- L. Pietronero; E. Tosatti; V. Tosatti; A. Vespignani (2001). "Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf". Physica A. 293 (1–2): 297–304. arXiv:cond-mat/9808305. Bibcode:2001PhyA..293..297P. doi:10.1016/S0378-4371(00)00633-6.
- Formann, A. K. (2010). "The Newcomb–Benford law in its relation to some common distributions". PLOS ONE. 5 (5): e10541. Bibcode:2010PLoSO...510541F. doi:10.1371/journal.pone.0010541. PMC 2866333. PMID 20479878.
- Theodore P. Hill (July–August 1998). "The first digit phenomenon" (PDF). American Scientist. 86 (4): 358. Bibcode:1998AmSci..86..358H. doi:10.1511/1998.4.358.
- Janvresse, Élise; Thierry (2004). "From Uniform Distributions to Benford's Law" (PDF). Journal of Applied Probability. 41 (4): 1203–1210. doi:10.1239/jap/1101840566. MR 2122815. Archived from the original (PDF) on 4 March 2016. Retrieved 13 August 2015.
- Pinkham, Roger S. (1961). "On the Distribution of First Significant Digits". Ann. Math. Statist. 32 (4): 1223–1230. doi:10.1214/aoms/1177704862.
- Weisstein, Eric W. "Benford's Law". mathworld.wolfram.com.
- Jamain, Adrien (September 2001). "Benford's Law" (PDF). Imperial College of London. Archived (PDF) from the original on 9 October 2022. Retrieved 15 November 2020.
- Berger, Arno (June 2011). "A basic theory of Benford's Law". Probability Surveys. 8 (2011): 1–126.
- Varian, Hal (1972). "Benford's Law (Letters to the Editor)". The American Statistician. 26 (3): 65. doi:10.1080/00031305.1972.10478934.
- "From Benford to Erdös". Radio Lab. Episode 2009-10-09. 30 September 2009.
- Walter R. Mebane, Jr., "Election Forensics: Vote Counts and Benford’s Law" (July 18, 2006).
- "Election forensics", The Economist (February 22, 2007).
- Deckert, Joseph; Myagkov, Mikhail; Ordeshook, Peter C. (2011). "Benford's Law and the Detection of Election Fraud". Political Analysis. 19 (3): 245–268. doi:10.1093/pan/mpr014. ISSN 1047-1987.
- Mebane, Walter R. (2011). "Comment on "Benford's Law and the Detection of Election Fraud"". Political Analysis. 19 (3): 269–272. doi:10.1093/pan/mpr024.
- Stephen Battersby Statistics hint at fraud in Iranian election New Scientist 24 June 2009
- Walter R. Mebane, Jr., "Note on the presidential election in Iran, June 2009" (University of Michigan, June 29, 2009), pp. 22–23.
- Roukema, Boudewijn F. (2014). "A first-digit anomaly in the 2009 Iranian presidential election". Journal of Applied Statistics. 41: 164–199. arXiv:0906.2789. Bibcode:2014JApS...41..164R. doi:10.1080/02664763.2013.838664. S2CID 88519550.
- Bernd Beber and Alexandra Scacco, "The Devil Is in the Digits: Evidence That Iran's Election Was Rigged", The Washington Post (June 20, 2009).
- Mark J. Nigrini, Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection (Hoboken, NJ: Wiley, 2012), pp. 132–35.
- Walter R. Mebane, Jr., "Election Forensics: The Second-Digit Benford's Law Test and Recent American Presidential Elections" in Election Fraud: Detecting and Deterring Electoral Manipulation, edited by R. Michael Alvarez et al. (Washington, D.C.: Brookings Institution Press, 2008), pp. 162–81. PDF
- Shikano, Susumu; Mack, Verena (2011). "When Does the Second-Digit Benford's Law-Test Signal an Election Fraud? Facts or Misleading Test Results". Jahrbücher für Nationalökonomie und Statistik. 231 (5–6): 719–732. doi:10.1515/jbnst-2011-5-610. S2CID 153896048.
- "Fact check: Deviation from Benford's Law does not prove election fraud". Reuters. 10 November 2020.
- Dacey, James (19 November 2020). "Benford's law and the 2020 US presidential election: nothing out of the ordinary". Physics World.
- William Goodman, The promises and pitfalls of Benford's law, Significance, Royal Statistical Society (June 2016), p. 38.
- Goldacre, Ben (16 September 2011). "The special trick that helps identify dodgy stats". The Guardian. Retrieved 1 February 2019.
- Sehity, Tarek el; Hoelzl, Erik; Kirchler, Erich (1 December 2005). "Price developments after a nominal shock: Benford's Law and psychological pricing after the euro introduction". International Journal of Research in Marketing. 22 (4): 471–480. doi:10.1016/j.ijresmar.2005.09.002. S2CID 154273305.
- Friar, JL; Goldman, T; Pérez-Mercader, J (2012). "Genome sizes and the benford distribution". PLOS ONE. 7 (5): e36624. arXiv:1205.6512. Bibcode:2012PLoSO...736624F. doi:10.1371/journal.pone.0036624. PMC 3356352. PMID 22629319.
- Diekmann, A (2007). "Not the First Digit! Using Benford's Law to detect fraudulent scientific data". J Appl Stat. 34 (3): 321–329. doi:10.1080/02664760601004940. hdl:20.500.11850/310246. S2CID 117402608.
- Stephens, M. A. (1970). "Use of the Kolmogorov–Smirnov, Cramér–von Mises and related statistics without extensive tables". Journal of the Royal Statistical Society, Series B. 32 (1): 115–122.
- Morrow, John (August 2014). Benford's Law, families of distributions and a test basis. London, UK. Retrieved 11 March 2022.
- Leemis, L. M.; Schmeiser, B. W.; Evans, D. L. (2000). "Survival distributions satisfying Benford's Law". The American Statistician. 54 (4): 236–241. doi:10.1080/00031305.2000.10474554. S2CID 122607770.
- Cho, W. K. T.; Gaines, B. J. (2007). "Breaking the (Benford) law: Statistical fraud detection in campaign finance". The American Statistician. 61 (3): 218–223. doi:10.1198/000313007X223496. S2CID 7938920.
- Suh, I. S.; Headrick, T. C.; Minaburo, S. (2011). "An effective and efficient analytic technique: A bootstrap regression procedure and Benford's Law". J. Forensic & Investigative Accounting. 3 (3).
- Ostrovski, Vladimir (May 2017). "Testing equivalence of multinomial distributions". Statistics & Probability Letters. 124: 77–82. doi:10.1016/j.spl.2017.01.004. S2CID 126293429.
- Washington, L. C. (1981). "Benford's Law for Fibonacci and Lucas Numbers". The Fibonacci Quarterly. 19 (2): 175–177.
- Duncan, R. L. (1967). "An Application of Uniform Distribution to the Fibonacci Numbers". The Fibonacci Quarterly. 5: 137–140.
- Sarkar, P. B. (1973). "An Observation on the Significant Digits of Binomial Coefficients and Factorials". Sankhya B. 35: 363–364.
- In general, the sequence k1, k2, k3, etc., satisfies Benford's law exactly, under the condition that log10 k is an irrational number. This is a straightforward consequence of the equidistribution theorem.
- Raimi, Ralph A. (August–September 1976). "The first digit problem". American Mathematical Monthly. 83 (7): 521–538. doi:10.2307/2319349. JSTOR 2319349.
- Zyga, Lisa; Phys.org. "New Pattern Found in Prime Numbers". phys.org. Retrieved 23 January 2022.
- Cho, Wendy K. Tam; Gaines, Brian J. (2007). "Breaking the (Benford) Law: Statistical Fraud Detection in Campaign Finance". The American Statistician. 61 (3): 218–223. doi:10.1198/000313007X223496. ISSN 0003-1305. JSTOR 27643897. S2CID 7938920. Retrieved 8 March 2022.
- Beer, Trevor W. (2009). "Terminal digit preference: beware of Benford's law". J. Clin. Pathol. 62 (2): 192. doi:10.1136/jcp.2008.061721. PMID 19181640. S2CID 206987736.
- Singleton, Tommie W. (May 1, 2011). "Understanding and Applying Benford’s Law", ISACA Journal, Information Systems Audit and Control Association. Retrieved Nov. 9, 2020.
- Durtschi, C; Hillison, W; Pacini, C (2004). "The effective use of Benford's law to assist in detecting fraud in accounting data". J Forensic Accounting. 5: 17–34.
- Dümbgen, L; Leuenberger, C (2008). "Explicit bounds for the approximation error in Benford's Law". Electronic Communications in Probability. 13: 99–112. arXiv:0705.4488. doi:10.1214/ECP.v13-1358. S2CID 2596996.
- Hill, Theodore P. (1995). "The Significant-Digit Phenomenon". The American Mathematical Monthly. 102 (4): 322–327. doi:10.1080/00029890.1995.11990578. JSTOR 2974952.
- Scott, P.D.; Fasli, M. (2001) "Benford's Law: An empirical investigation and a novel explanation" Archived 13 December 2014 at the Wayback Machine. CSM Technical Report 349, Department of Computer Science, Univ. Essex
- Suh, I. S.; Headrick, T. C. (2010). "A comparative analysis of the bootstrap versus traditional statistical procedures applied to digital analysis based on Benford's law" (PDF). Journal of Forensic and Investigative Accounting. 2 (2): 144–175.
- Arno Berger; Theodore P. Hill (2017). "What is...Benford's law?" (PDF). Notices of the AMS. 64 (2): 132–134. doi:10.1090/noti1477.
- Arno Berger & Theodore P. Hill (2015). An Introduction to Benford's Law. Princeton University Press. ISBN 978-0-691-16306-2.
- Alex Ely Kossovsky. Benford's Law: Theory, the General Law of Relative Quantities, and Forensic Fraud Detection Applications, 2014, World Scientific Publishing. ISBN 978-981-4583-68-8.
- "Benford's Law – Wolfram MathWorld". Mathworld.wolfram.com. 14 June 2012. Retrieved 26 June 2012.
- Alessandro Gambini; et al. (2012). "Probability of digits by dividing random numbers: A ψ and ζ functions approach" (PDF). Expositiones Mathematicae. 30 (4): 223–238. doi:10.1016/j.exmath.2012.03.001.
- Sehity; Hoelzl, Erik; Kirchler, Erich (2005). "Price developments after a nominal shock: Benford's law and psychological pricing after the euro introduction". International Journal of Research in Marketing. 22 (4): 471–480. doi:10.1016/j.ijresmar.2005.09.002. S2CID 154273305.
- Nicolas Gauvrit; Jean-Paul Delahaye (2011). Scatter and regularity implies Benford's law...and more. Zenil: Randomness Through Computation: Some Answers, More Questions. pp. 58–69. arXiv:0910.1359. Bibcode:2009arXiv0910.1359G. doi:10.1142/9789814327756_0004. ISBN 978-9814327756. S2CID 88518074.
- Bernhard Rauch; Max Göttsche; Gernot Brähler; Stefan Engel (August 2011). "Fact and Fiction in EU-Governmental Economic Data". German Economic Review. 12 (3): 243–255. doi:10.1111/j.1468-0475.2011.00542.x. S2CID 155072460.
- Wendy Cho & Brian Gaines (August 2007). "Breaking the (Benford) Law: statistical fraud detection in campaign finance". The American Statistician. 61 (3): 218–223. doi:10.1198/000313007X223496. S2CID 7938920.
- Geiringer, Hilda; Furlan, L. V. (1948). "The Law of Harmony in Statistics: An Investigation of the Metrical Interdependence of Social Phenomena. by L. V. Furlan". Journal of the American Statistical Association. 43 (242): 325–328. doi:10.2307/2280379. JSTOR 2280379.